Time Travel and Computing

Hans Moravec

1991

The last few years have been good for time machines. Kip Thorne's
renowned general relativity group at Caltech invented a new quantum
gravitational approach to building a time gate, and, in an
international collaboration, gave a plausible rebuttal of "grandfather
paradox" arguments against time travel. Another respected group
suggested time machines that exploit quantum mechanical time
uncertainty. The technical requirements for these suggestions exceed
our present capabilities, but each new approach seems less onerous
than the last. There is hope yet that time travel will eventually
become possible, even cheap.

Most time-machine fiction deals with the sociological implications of
temporal trips or messages--indeed the time travel is often a mere
literary device for placing humans in unusual situations. A recent
paper from Thorne's group, by contrast, examines time travel at a
basic physical level, deriving the quantum mechanical wave functions
of systems that contain temporal loops. This article looks at the
situation at an intermediate scale--the uses of time travel, packaged
as negative time delay elements, on computation. This view is
interesting because, on one hand, it predicts colossal improvements in
the ability to solve important problems, and, on the other, provides a
crisp logical metaphor for macroscopic implications of time
travel.

A Brief History of Time Travel

When H.G. Wells wrote The Time Machine , his first novel,
in 1895, the scientific world was unimpressed. In that Victorian age
Science was in the process of crossing the t's and dotting the i's on
Newtonian mechanics, and worrying about unemployment in the coming age
of fully codified physical knowledge. Time was a rigid universal
framework for the clockwork processes of physical law, without the
slightest hint of a mechanism for evading its unvarying progress.
That time machines were provably impossible was conveyed convincingly
to generations of students, and echoes to this day.

The physics revolution at the beginning of the twentieth century
shattered objective certainty about the immutability of time, but not
most physicists' gut assurances. Special relativity combined space
and time into a single continuum, with velocity a kind of rotation
that transformed one into the other. A barrier--the speed of
light--still separated the "spacelike" from the "timelike", but it was
a fragile one. There was nothing in special relativity itself to
preclude the existence of particles, now dubbed tachyons, that always
moved faster than light. A tachyon message returned by a distant,
rapidly receding relay could arrive before it was sent, a consequence
construed by the conservative majority of the physics community as an
indictment of tachyons. The impossibility of time travel (or closed
causal loops, or, in the language of relativity, closed "timelike"
loops) had become an axiom of physical law. Tachyons have not, in
fact, been detected, even though they should be creatable with
arbitrarily little energy (the faster they move, the less it takes),
so perhaps the conservative majority is correct in this case (but
perhaps they just have a tendency to hide - see below).

General relativity patches together tiny regions of flat
special-relativistic spacetime into large gravity-warped structures.
Powerful gravity fields imply radically convoluted spacetimes. Kurt
Gödel was first, in 1949, to notice that general relativity
predicted time travel under certain circumstances. In Gödel's
solution to Einstein's equations, the centrifugal tendency of a
rotating universe exactly balances its tendency to gravitationally
collapse. In such a universe the spacelike and timelike directions
are skewed sufficiently that a spaceship accelerating around the
universe can arrange to return to the place and time of
its launch, giving the crew an opportunity to wave bon
voyage to their departing younger selves. General relativity
has been repeatedly confirmed experimentally in the large scale, so
those who dislike the prediction take solace in the fact that our
universe appears hardly to rotate.

The next major class of solutions, made in the 1960s by Roy Kerr and
Ezra Newman and colleagues, are harder to dismiss. The Kerr-Newman
solutions of the Einstein equations are for rapidly rotating and/or
charged black holes. In the most extreme of these, the rotation of
the body counteracts the gravitation enough to expose the twisted
viscera of the black hole (normally hidden behind a discreet, one way,
event horizon) including regions of negative spacetime,
from which a spaceship could return to the outside universe before it
entered. Attempts by the conservative majority to find independent
reasons for a cosmic censorship rule to prevent such lewd
exposure have been unsuccessful, so far. Their only comfort is that
it would take about the mass of a galaxy, with extraordinary spin, to
make a practical time machine this way.

In 1974 Frank Tipler published another solution to the general
relativity equations, this time for the region around an extremely
dense, very long rapidly spinning cylinder--dense as a neutron star,
the diameter of a city block, with its surface moving at about one
fourth the speed of light, and infinitely long, because that
simplified the mathematics. Spacetime wraps itself around such an
object like a roll of paper, producing alternate layers of negative
and positive spacetime. A carefully aimed spaceship could swing
around such an object, staying mostly in a negative region, and come
out before it left. A finite length cylinder should also work, and
might allow a time machine with only the mass of a star. But, the
conservatives say, maybe it's not possible to prevent a finite
cylinder from gravitationally collapsing lengthwise.

As yet, no one has devised a satisfactory comprehensive single theory
that combines gravity and quantum mechanics--many try, and the
implications of such a theory promise to be awesome. In 1988 Kip
Thorne and company, patching together partial theories, described a
time machine using both quantum mechanics and general relativity. A
tiny, spontaneously formed, gravitational spacetime wormhole is pulled
out of the hyperactive froth that is the quantum vacuum, and
stabilized, by two large conductive plates, resembling an electrical
capacitor. Initially these plates are as closely spaced as possible,
and each becomes host to one "mouth" of the wormhole. When they are
separated in our normal spacetime, they yet remain connected through
the wormhole, which is an independent spacetime tunnel. Regardless of
their external separation, a message or object entering one mouth
appears instantly (by its own reckoning) at the other, as if the
mouths were the two sides of a single door. Thorne's group then uses
special relativity to differentially age the two mouths. One is taken
on a "twin paradox" round trip at near the speed of light, so that
less time elapses for it than its stationary counterpart. When it
returns, the external separation between the two mouths has a time as
well as a space component. A message sent into the itinerant mouth
exits from the stationary one after a delay. And a message delivered
into the stationary mouth exits from the traveller before it was sent!
This kind of machine could perhaps be constructed with a planet's
worth of aluminum spread out into plates the area of Earth's orbit,
separated by the size of an atom--still beyond our means, but getting
closer. The non-linear equations of general relativity are
notoriously hard to solve, and only the very simplest cases have been
explored. Even more significantly, there isn't any theory of quantum
gravity yet at all. It's quite possible that in all this unexplored
territory, waiting to be discovered, lie quite feasible ways to build
time machines.

Another approach to time travel asks why it isn't observed routinely.
There is no intrinsic time direction in Newton's mechanics nor in the
differential equations of the new physics. The future determines the
past just as fully as the past determines the future. Why, then, can
our past selves leave messages for our future, but seemingly
absolutely never the other way around? This question has not been
definitively answered. Attempted explanations involve "boundary
conditions", the initial values of physical quantities at the edges of
space and time, which the differential equations of physical law then
fill out. The universe must be somehow very different at one end than
the other, and this difference orients the arrow of time. Most common
is the thermodynamic explanation, which talks about the state of
disorder or "entropy" of matter and energy. The universe started in a
very rare, highly ordered state, and is running down into increasingly
common states of disorder. Though this explains why a ship can't run
its engines by separating water into steam and ice, it does not
explain why one can't send today's lottery numbers into yesterday by
expending a few megawatt hours of energy.

One explanation which does was offered by John Wheeler
and Richard Feynman in 1945. They noted that Maxwell's equations, the
first "modern" physical theory, give two solutions for the effect of
accelerating an electric charge. One, called the retarded wave,
follows the acceleration and describes an electromagnetic disturbance
diverging outward at the speed of light--the radio waves which link
our civilization. The other, called the advanced wave, is for a
similar disturbance that precedes the acceleration and converges on it
(or, in another way of looking at it, diverges backwards in time).
This latter wave is never seen. Wheeler and Feynman's analysis
assumes that the advanced wave is, in fact, produced, and expands
outwards into the past. There, eventually, it encounters a condition,
perhaps the extreme density of the big bang at the beginning of the
universe, that reflects it, producing a retarded wave exactly out of
phase, that retraces its spacetime and exactly cancels it out. The
retarded wave from an accelerated charge is not cancelled in an
analogous manner, because there is no reflector in the future of the
universe to reverse it--perhaps because the universe is "open", and
expands forever. If this is a correct explanation, it might be
possible to send signals backward in time by means of a kind a
reflector that acts like the big bang--maybe a black hole.

Such a reflector would reverse and return a retarded wave, cancelling
and thus apparently preventing it. If the reflector were installed
one light year away from a light source aimed at it, it would suppress
light from the source one year before installation.
Similarly the suppression would go away one year before the reflector
was removed. Messages could be sent a year into the past simply by
moving the reflector in and out of the beam. Recently John Cramer
devised an interpretation of Quantum Mechanics, called the
transactional model, that uses this approach to explain every
interaction. Transmission of a photon from one place to another
actually involves two signals, one moving forward in time, the other
backward, a "handshake", between the two locations. In Cramer's model
a time communicator could be built that works much like the
Wheeler-Feynman type, but using a wave absorber to prevent a
transaction, rather than a reflector to cancel it.

There is a spookier possibility. Suppose it is easy to send messages
to the past, but that forward causality also holds (i.e. past events
determine the future). In one way of reasoning about it, a message
sent to the past will "alter" the entire history following its
receipt, including the event that sent it, and thus the message
itself. Thus altered, the message will change the past in a different
way, and so on, until some "equilibrium" is reached--the simplest
being the situation where no message at all is sent. Time travel may
thus act to erase itself (an idea Larry Niven fans will recognize as
"Niven's Law"). This situation can be modeled quantum mechanically.
If the message is a particle sent to the past, the wave function for
that particle will subsequently propagate into the future, where it
encounters and "interferes" with its original self, cancelling or
reinforcing depending on the relative phase (which depends on the
round trip length and other things). Now, the resultant wave function
indicates the possible places the particle might actually be found in
a measurement--large magnitudes are likely locations, near zero values
unlikely ones. A round trip that causes the particle wave function to
be cancel itself means that a particle is unlikely ever to be
positioned to start the trip. It is exactly this kind of quantum
probability effect (without reference to time travel) that powerfully
confines electrons to discrete shells about the atomic nucleus and
causes the "Pauli exclusion principle" that prevents two
indistinguishable electrons from being in the same place
simultaneously.

The Wheeler-Feynman advanced wave theory, Cramer's transactional model
of quantum mechanics, and even quantum electrodynamics, the most
accurately verified physical theory we have, all involve interactions
that reach backwards as well as forwards in time. It may well be that
time travel is as common as dirt, and shapes our physical laws, but
conspires, by wave function interference, to prevent any operations
that would result in logically inconsistent situations, and so makes
overt macroscopic time travel difficult. Even with a time machine you
will never succeed in preventing your own birth, or changing the
antecedents of any present observation--some odd co-incidence or
accident (perhaps one disabling your time machine) will always thwart
the attempt.

But this does not rule out carefully contrived logically
consistent causal loops (The recent paper by Friedman, Morris,
Novikov, Escheverria, Klinkhammer, Thorne and Yurtsever examines the
consequences of such constraints on the wormhole time machine
mentioned earlier). Wave interference typically appears as banded
patterns with a strong central (zero order) peak surrounded by a dark
fringe, itself surrounded by the another (first order) bright fringe,
and so on. If causal loops behave this way, it will be necessary to
make major perturbations in experimental arrangement to skip from the
usual zero order (no overt time travel) situation to a first order
case of a non trivial consistent causal loop.

Let's suppose that eventually some approach results in a practical,
even cheap, way to package time machines to make negative time
delay elements --which output signals that predict what their
inputs will receive some fixed time later. Such devices would have
very interesting consequences for computational problems, which
increasingly means almost every field of activity.

Time Loop Logic

Computer circuitry is composed of devices called gates that combine
binary signals to produce other such signals. The simplest gate is an
amplifier whose output is identical to its input. Almost as simple is
the NOT gate, or inverter, whose output is 0 when its input is 1, and
vice versa. All gates take a little time to respond to changes in
their input--typically a few billionths of a second. When the input of
an amplifier is connected to its output, the circuit will lock up with
its output permanently at either 0 or 1. A NOT gate in a similar loop
tends to oscillate rapidly between 0 and 1, at a rate that depends on
its delay. It is possible to slow down this oscillation by inserting
extra delay into the loop. Conversely, a negative delay element would
speed up the oscillation. Imagine the two cases of an amplifier and a
NOT gate with a certain delays each with its output connected to its
input through a negative delay device:

Causal loops involving an amplifier (triangle) and inverter (triangle
with circle) in causal loops with negative time delays (- t) that
cancel their forward delay.

The amplifier circuit is in a consistent causal loop--when first
switched on, it can permanently assume either 0 or 1 without
contradiction. The loop with the inverter, on the other hand, is a
simple case of the classical time travel grandfather paradox, a
paradoxical causal loop. An input of 1 to the inverter gives an
output of 0, which is brought back in time to contradict the input.
It takes some quantum mechanics to make sense of the situation, and we
will have to say something about how the signals are physically
represented. Most digital circuits represent signals as electrical
voltages or currents in wires, which is inconvenient because electrons
interact with each other and with matter in complex, hard to analyze
ways. Some experimental circuits use much simpler space-crossing
beams of light . Let's suppose 1 and 0 are encoded as coherent light
beams of opposite phase (perfectly out of step with one another--one
crests where the other has troughs). In that case a 0 that meets
meets a 1, as in the inverter circuit, will simply cancel . Either
alternative would have zero net probability, and the circuit (perhaps
containing a charged laser, ready to emit a beam) should simply fail
to turn on (ignite) at all, somewhat like a ball balanced on a knife
edge that, against all odds, teeters indefinitely instead of falling
to one side or the other. This is Niven's law at work in the small.
The circuit finds itself perpetually in a dark fringe of an
interference pattern.

On the other hand, suppose the two signals are encoded as beams of
identical phase but opposite polarization (horizontal or vertical
direction of vibration). Then the light in the inverter circuit will
be in step with itself, and so should ignite. But what about the
polarization? Neither of the possibilities individually results in a
consistent situation, but quantum mechanics allows these to overlap in
a superposition. The superposition that contains equal
measures of 0 and 1 is changed by the inverter to an indistinguishable
superposition of 1 and 0, and so allows the circuit to remain
consistent. Light with equal amounts of vertical and horizontal
polarization is unpolarized . The inverting causal loop
insures that the light remains perfectly unpolarized. An attempt to
change this, for instance by slipping a polarizing filter somewhere
into the circuit, should cause the beam to extinguish, as for phase
modulation.

Imagine that the beam in the polarization modulated inverter circuit
is sampled, perhaps by a partially silvered mirror, and examined.
Quantum mechanics tells us that each extracted photon will be in a
superposition of polarization until the polarization is measured, at
which point it "collapses" to one possibility or the other. Sometimes
the photon is in a 1 state, equally often it is found to be a 0. But
suppose, as in existing computer circuitry, signals are represented
not by single particles but by bulk aggregates which behave almost
like classical continuous variables. An aggregate of random values
will act like a signal intermediate in value between 0 and 1, say
around 1/2. In classical terms, it makes sense that a circuit that
converts 0 to 1 and 1 to 0 should leave some intermediate value
unchanged:

There are other ways to view the situation. An inverting loop without
a negative delay will oscillate between 0 and 1 at a frequency that
depends on the time delay of the inverter. As we insert increasingly
long negative delays in the circuit, the frequency of the oscillation
increases. The superposition is when the frequency reaches infinity,
and the circuit is in the 0 and 1 states simultaneously. Note also
that the the inverter's 1/2 state is extremely unlikely--a small
deviation in the input would cause the output to saturate at one of
the two extremes. Yet the circuit indefinitely maintains precisely
that unlikely situation. The negative delay connects regions of
spacetime in the same way that the orbit of an electron around an
atomic nucleus connects different parts of its wave function, and so
creates electron shells separated by empty regions. Like a focussing
lens that brings to a single spot photons that would otherwise have
landed all over a target, it produces interference patterns that make
a few states of the world very likely and and the rest unlikely. It's
just that, by eliminating the standard possibilities, negative time
delays make likely things that are otherwise nearly
impossible.

Infinite Iteration

Here's a more complicated case. Make a computing box that accepts an
input, which represents an approximate solution to some problem, and
produces an output that is an improved approximation. Conventionally
you would apply such a computation repeatedly a finite number of
times, and then settle for the better, but still approximate, result.
Given an appropriate negative delay something else is
possible:

In this arrangement the result of each iteration of the function is
brought back in time to serve as the "first" approximation. As soon
as the machine is activated, a so-called "fixed-point" of F, an input
which produces an identical output, usually signaling a perfect
answer, appears (by an extraordinary coincidence!) immediately and
steadily, just as either 1 or 0 appears in the simple amplifier
loop. If the iteration does not converge, that is, if F has no fixed
point, the computer outputs and inputs will shut down or hover in an
unlikely intermediate state, like the inverting loop.

The Compleat NP Machine

Generally speaking, whether a problem can be solved in a computer
depends on its computational complexity, which describes
how the difficulty grows as the problem increases in size. Finding
the largest number in a list of numbers takes time proportional to the
list length. A graph of computing time versus list length yields a
straight line--the complexity is linear , and this is an
easy problem. Sorting the list into numerical order is harder.
Simple methods take time proportional to the square of the list
length--the graph is an upward curving parabola--and even the best
possible sorting methods give slight upward curves. The cost of
solving a system of n equations in n
unknowns grows as the cube of n , and other problems grow
as the fourth, fifth or even higher powers of the problem size. But
any problem whose difficulty can be expressed as a fixed power of size
is of polynomial complexity, and even large instances are
soon solvable in a world where computer power doubles every few years.

Not so problems of exponential complexity. These
multiply in cost with each fixed increment of problem size. Whereas a
linear problem grows as 1, 2, 3, 4, 5, 6, ..., and a cubic as 1, 8,
27, 64, 125, 216, ..., an exponential problem's cost may grow as 1,
10, 100, 1000, 10000, 100000 and so on to the astronomical. An
important class of apparently exponential problems are called NP,
short for Nondeterministic Polynomial, meaning exponential problems
that could be solved in polynomial time given an exponential supply of
computers to examine alternatives. Examples include many design
problems such as finding the best arrangement of logic gates, or the
fastest program, to compute a given function, and also finding limited
length proofs of mathematical theorems. Solving such problems could
synergistically increase the power of the machines that solve
them. Even poetry or music writing might fit, if approached as the
problem of finding the best sequence of constrained words or sounds to
express an idea or a state of mind.

The hard core of NP problems are called NP complete, and it 's been
shown that a fast solution for any NP complete problem can be
translated into a fast solution for any other. One famous and
convenient NP task is the traveling salesman problem
--given n cities and the distances between pairs of them,
find the shortest tour that passes through each city exactly
once.

A version of the time loop iteration box of the last section can
quickly solve such a task. The F box for this problem takes as input
a particular tour, that is, a permutation of the cities. It also has
a knob whose position specifies a limit on the length of the tour.
The box calculates the input tour's length, and outputs the same tour
if that length is less than or equal to the limit. If the length
exceeds the limit, the box generates the next permutation (by some
permutation counting scheme). When the circuit is activated it
settles down immediately to a tour shorter than the limit, or sticks
in an undecided state if no such solution exists--i.e. if the limit
was set too low.

Turning the knob to find the boundary between these two conditions
solves the problem. The "knob turning" can, of course, be done
automatically by additional machinery, for instance an ordinary
computer. In fact, one can imagine building a very general
"chronocomputer" by simply slipping a sufficiently large "negative
delay register board" into a peripheral slot of any computer.

Solving Chess

NP problems quickly exceed conventional computer capacity, but of
exponential problems they are the easiest to solve. In an NP problem
there are an exponential number of candidate solutions to be
considered, but the correctness and cost of each candidate (a salesman
tour, a circuit for a function, a mathematical proof ) can be checked
easily, a fact our time-loop NP computer exploits. But there exist
harder exponential problems where evaluating a single candidate answer
is itself an exponential problem.

Finding the best move in a game like chess is an instance of such a
problem. Your best move can be found by considering all possibilities
for your move, then for each of those, all possible responses by the
opponent, and all your possible responses to all of those, and so on,
until a "tree" of all possible games has been mapped out. Then,
working backwards from the "leaves" of this tree, i.e. the final moves
of the games, all but the best moves can be pruned away. It is easy
to evaluate possible last moves--from the point of view of the player
whose turn it is, some are wins, some are losses and some result in
draws, and no finer distinctions need be made. When all but the best
last moves are discarded, the relative merits of second-to-last moves
become apparent--each is given by the best last move that results from
it. This pruning proceeds to earlier and earlier moves until the
beginning of the game, and the best first move, is met. The chain of
best moves from first to last is a perfect game of chess.

One can imagine a kind of generalized chess played on a variety of
board sizes, with 8 by 8 giving the standard game. Finding a perfect
game is easy on small boards, with few possible moves, but becomes
enormously difficult as the board grows. The standard game is already
so difficult that, despite known mathematical shortcuts, there is no
hope that a conventional computer, even one using all the time and
matter in the universe, could search the entire game tree. Today's
chess computers examine a few levels of the tree, and use a formula to
(very imperfectly) guess the value of the rest.

But time loop computers seem to greatly transcend the power of
conventional machines. Can the approach used for NP problems be
applied here? If, instead of city tours, the input to the computation
box is the move sequence for an entire game, the box could be arranged
to stabilize only on sufficiently "good" games. But, unlike traveling
salesman tours, games have only three values--win, lose or draw (for
white, say). There seems no easy way to assign a number to indicate
how nearly perfect a game is--and a perfect game itself is recognized
only by considering and eliminating all other possible games.

There may be a devious way, however, to use negative delays to fold
the massive tree search in time, in a fashion that makes the NP solver
look positively pedestrian. This construction is going to resemble a
mathematical induction. Suppose we have a circuit (call it the
n box) that, given a particular chessboard position
n moves into a game, is able to immediately tell us its
value (win, lose or draw), as if it had searched the entire move tree
from there. We could then build a box that provides the same services
for a position n -1 moves into the game by adding a
"single move unit" that takes the n -1 board, and, one by
one, generates the possible next moves (typically about 30 of them),
and feeds the resulting boards to the n box. It compares
the n box results to one another, and chooses the best
(for the player whose move it is). The time taken to do this is
cancelled by a negative time delay element, and so the n
-1 box, like the n box, produces the value of a position
as soon as the position is presented:

The n box is itself composed of a single move unit and an
n +1 box, and so on. The chain can be stopped eventually
because each single move unit has a "short circuit" path that it uses
when it encounters a position that, because of a king capture or
excessive move count, is already a win, loss, or draw and so requires
no further search. The whole machine is simply a chain of single move
units and negative time delays as long as the longest game (a few
hundred moves for standard chess), with the last unit set to
permanently return a draw, signifying a stalemate.

Feed it the standard starting chess setup and the machine will
immediately indicate whether the game is a forced win for white, black
or a draw. To play (perfect) chess, give the machine the result of
each of your possible moves, and choose one it finds best. But what
is the machine up to in these deliberations? Having accepted time
travel logic, the operation of the first single move unit is easy to
intuit--it simply evaluates a few dozen possibilities and returns the
answer, through the agency of a negative time delay, before its
deliberations are complete. The second stage is more mysterious. Its
input is changed by the first unit after it has delivered an answer,
but while it is still doing the calculations for that answer! It is
being asked to do several dozen different computations simultaneously.
The third and subsequent units are required to be in even more
bewildering states of superposition. The machine would certainly not
work if its interior were being observed, as quantum observation
collapses states of superposition to particular possibilities. Each
single move unit would be seen to receive new boards to evaluate
before it had completed the previous ones. Even weak observations
(like hearing a whisper through a heavy wall), or indirect or delayed
ones, would cause occasional collapses. To have any chance of giving
correct answers, the machine must be totally protected from any
peeking, perhaps by burying it in many layers of exotic shielding, or
by placing it out of range of observation. Even then, its operation
depends on aspects of reality hidden from present day experiments,
such as the actual independent existence of the alternative worlds
that appear in quantum mechanical calculations. Time will
tell.

The non Computable

Exponential problems are big, but finite. Our universe may not be
large or long lasting enough to solve serious instances of them
without time travel, but it is possible to conceive of universes that
are. But some problems are so hard that no bounded universe would be
sufficient--their complexity is infinite. Kurt Gödel, who
discovered rotating-universe time travel in general relativity, is
much better known for shocking the mathematical community with his
incompleteness theorems, showing that in any consistent and
sufficiently interesting mathematical theory, there are unverifiable
truths.

In the seventeenth century Pierre Fermat wrote that he had discovered
a truly remarkable proof that there are no integer solutions to the
equation Xn + Yn = Zn when X, Y, Z > 1 and n > 2, but
unfortunately the margin of the book he was annotating was too small
to contain it. Subsequent generations of mathematicians have searched
in vain for this (or any) proof to "Fermat's last theorem". Nor has a
counterexample been found to prove it wrong. Most probably there was
a subtle, unnoticed, flaw in Fermat's "proof". Today the theorem
remains a good candidate for a Gödelian true but unprovable
statement in arithmetic. If so, we won't be able to prove that it is
unprovable, since such a proof would imply that no counterexample
(providing a negative proof of the theorem) can be found, thus proving
that no such counterexample exists, thus proving that the theorem is
true, contradicting its unprovability.

But Gödel's theorems hinge on the finiteness of proofs. We
might try to evade them with time loops. Suppose, as in the NP
machine example, we build an F box that
takes as input a quartet of numbers {X,Y,Z,n} and tests them to see if
they constitute a counterexample to Fermat's conjecture. If so, the
same numbers are presented at the output of the box. If not, the
"next" quartet is generated and fed via a negative time delay back to
the input. When the device is switched on, it immediately shows a
counterexample to the conjecture, if such exists, proving the
conjecture false. If , on the other hand, the box's signals hover in
an intermediate state, the theorem must be true.

There's a problem, of course. A finitely sized machine can examine
numbers only as wide as its signal paths. The time loop gives a lot
of leverage, since, in an instant, it examines a number of cases
exponential in the number of digits in these paths. But the number is
still finite, and if the machine fails to find a counterexample, one
can't be sure there isn't one that just happens to exceed its
capacity. A search for any other kind of proof or disproof founders
on this same Gödelian impasse--though a time loop machine can
examine all alternatives of a given length, a wonderful proof may yet
elude us because the machine we've built is too small to contain
it.

In situations like Fermat's last theorem we may willing to forgo an
astronomically (or infinitely) large proof or counterexample if we are
told by a trustworthy source that the statement is true or false--a
compact answer that fits in a very small machine. The time computer
techniques we've discussed thus far examine, in a short fixed period,
a number of cases exponential in the size of the machine. It may be
possible to substitute time for space, to build a small, continuously
operating machine that examines an exponentially increasing number of
cases as time passes. Take, for instance, a single move unit from the
chess machine example, and connect its output to its input via a
negative time delay. Observed, the system acts as a simple time loop,
as in the NP example. But unobserved, the move sequencing might spawn
multiple alternatives, just as in the chess solver, but of the
original unit itself rather than units downstream. As time goes by,
each copy repeatedly splits itself again, and the number of parallel
machines grows exponentially. After a few hundred iterations, this
single unit will have covered the same tree of alternative games as
the extended chess machine. But by letting it run beyond that point,
longer games can be examined. The answer is found by the peculiar
procedure of inputting the initial chess board to the single move unit
and peeking at its output value at the start of the computation, then
tightly closing an observation-proof box around it. The answer you
receive right away is trustworthy only if the machine is subsequently
left to run, undisturbed and unobserved, for long enough .

For more general problems, let's consider an abstract device conceived
in the 1930s by British mathematician Alan Turing, inspired by David
Hilbert's concept of "mechanizing" mathematics. A Turing Machine is a
finite, usually simple, computer or "finite state machine" connected
to a "read/write head" that moves back and forth on an indefinitely
long tape. The machine proceeds in regular steps, at each reading a
symbol on the square of the tape under the head, then, controlled by
an unambiguous finite internal list of rules, writing a symbol in its
place, and moving one square forward or back. By convention, the
internal rulebook for any given Turing Machine is fixed, but its tape
can be initialized with an arbitrary string of symbols. Turing showed
that such machines, generally using lots of tape to store and
reference initial inputs, intermediate results and final answers,
existed for every conceivable computation:

Turing also showed that there exist machines that interpret the
initial contents of their tape as a rulebook of another Turing
Machine, and proceed, slowly, to simulate this other machine. These
are called "Universal" Turing Machines, for their ability to do any
computation that another machine can do. A Universal Turing Machine
is a good mathematical model for a digital computer, and Turing used
the concept to prove "non-computability" theorems that are equivalent
to (but more straightforward than) Gödel's "unprovability"
results.

By appropriately initializing its tape, one can program a Universal
Turing Machine, like a conventional computer, to do just about any
computation, for instance to search through {X,Y,Z,n} quartets looking
for a counterexample to Fermat's Last Theorem, or through inference
chains for a proof or disproof. A literal Turing Machine, though, is
very slow--it spends most of its time crawling back and forth across
increasingly long stretches of tape to get at widely separated bits of
information. But it is amenable to a time-computer transformation.
Imagine that the tape runs not in space, but in time. At any instant
it is just a single square (probably a circuit) storing a symbol.
Instead of moving left or right, the head sends a message to its past
self through a negative delay, or to its future self through a
positive delay:

Many questions could be posed about this design. A few have answers.
Because the machine is created at a particular time, and does not
exist before then, the "tape" ends abruptly in the backwards
direction. This is not a problem because in any computation there are
straightforward ways to "fold" tape usage to exclusively the forward
portion. Interesting computations will use a lot of tape--i.e. will
extend far into the future. But if the program copies its ultimate
answer to the cell initially under the head, the answer will be
immediately available. As with the the folded chess machine, one must
start the "Temporal Turing Machine", quickly peek at its tape cell for
the answer, and then seal it up, to let the computation run
undisturbed, unobserved. But then how is the program installed on the
tape, on cells corresponding to time periods when the machine is
sealed? The easiest answer is to encode it in the finite state
portion of the machine rather than the tape.

In its sealed box, the head effectively runs forward and backward in
time, repeatedly overwriting cells, spawning multiple worlds ever
faster. Yet the number of situations the machine can examine
(different Fermat's Last Theorem quartets, for instance), can at most
grow exponentially in the machine's running time. This is truly
prodigious by conventional standards, but still quite finite. So here
we are, as far out on a speculative limb as I'm willing to climb, and
infinite problems remain hopelessly out of reach. It's time to
quit.

Time Travel, Consciousness and Reality

The above constructions are no less strange when applied to
"conventional" time-travel scenarios. Imagine that the computation
boxes actually contain human beings--a person can quite plausibly
evaluate the length of a traveling salesman tour, or enumerate the
possible next moves in a chess game. Assuming, as before, that all
observations must be logically consistent, what would a human, so
embedded in a time loop, experience? When the NP machine produces a
solution, its computation box evaluates only one case--a correct one,
since quantum interference cancels others. To the person in the
computation box, no less than to those outside, a correct answer
appears, as if by magic, on the box's input--found correct,
transcribed to the output, and relayed back in time, it also creates
that input--a circular but logically consistent situation. On the
other hand, there is little reason to believe that worlds containing
incorrect solutions can be experienced, since such experiences would
involve logical contradictions.

Things are not so straightforward for persons inside the chess
machine--let's put a friend in each single move unit. The opening
position unit feeds first moves one after another to the rest of
machine (the "2" box), and examines the reponses--no difficulty. But
the second unit is required to be in a superposition of states,
simultaneously examining each of the first move's alternatives. When
made of dumb machinery, it was reasonable to simply seal it off from
observation, to prevent spoiling the computation by collapsing the
superposition to a single possibility. But an intelligent friend will
probably need to be released sooner or later, and will retain a memory
of the computation. In the strange logic of quantum mechanics, this
memory, released from the box, constitutes an observation that
retroactively collapses the computation. We could, of course,
preserve the superposition and the computation by callously keeping
the boxes with our friends permanently sealed, or dropping them down a
black hole, or accelerating them away beyond observation. But then
there will be no way to learn about their experiences. So,
apparently, one can either exploit parallel worlds, or experience a
single one of them, but not both. "All the world's a stage" wrote
William Shakespeare, and apparently we players act in only one story
at a time. But when we close our eyes and listen closely, we hear
from the wings the echoes of other stories. Whether they really are
other plays (with audiences?), or just sound effects by a clever stage
manager, remains undecided.